Cement and Concrete Research 41 (2011) 1130–1142

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Cement and Concrete Research

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Influence of the meso-structure in dynamic fracture simulation of concrete undertensile loading

L. Snozzi a, A. Caballero b, J.F. Molinari a,⁎a LSMS, ENAC, Swiss Federal Institute of Technology (EPFL), CH-1015, Lausanne, Switzerlandb BBR VT International Ltd, Zürich, Switzerland

⁎ Corresponding author. Tel.: +41 216932411; fax: +E-mail address: jean-francois.molinari@epfl.ch (J.F. M

0008-8846/$ – see front matter © 2011 Elsevier Ltd. Aldoi:10.1016/j.cemconres.2011.06.016

a b s t r a c t

a r t i c l e i n f o

Article history:Received 8 September 2010Accepted 30 June 2011

Keywords:C. Finite element analysisC. Tensile propertiesC. MicromechanicsE. Concrete

We investigate the dynamic behavior of concrete in relation to its composition within a computationalframework (FEM). Concrete is modeled using a meso-mechanical approach in which aggregates and mortarare represented explicitly. Both continuum phases are considered to behave elastically, while nucleation,coalescence and propagation of cracks are modeled using the cohesive-element approach.In order to understand the loading-rate sensitivity of concrete, we simulate direct tensile-tests for strain ratesranging 1–1000 s−1. We investigate the influence of aggregate properties (internal ordering, size distributionand toughness) on peak strength and dissipated fracture energy. We show that a rate independentconstitutive law captures the general increase of peak strengthwith strain rate. However, a phenomenologicalrate-dependent cohesive law is needed to obtain a better agreement with experiments. Furthermore, at lowrates, peak strength is sensitive to the inclusions' toughness, while the matrix dominates the mechanicalbehavior at high rates.

41 216936340.olinari).

l rights reserved.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Structures made of concrete, a highly heterogeneous material, areoften subjected to dynamic loads. These may include impact,explosions and seismic loads. Although it is well known that themechanical behavior of concrete depends on strain rate, the ratedependency of this material under dynamic loading conditions isusually not taken into account in the design codes.

Among others, peak strength, strain at peak strength and totaldissipated fracture energy have been shown to be dependent on strainrate, [22,30]. In general, concrete exhibits higher peak strength whenit is loaded at higher strain rates and this increase has been shown togrow linearly with the strain rate. Fortunately, the augmentation ofstrength is accompanied by a larger dissipation of fracture energy, soconcrete does not become more brittle.

Nevertheless, the dynamic behavior of concrete under high-strain-rate tensile loading has still not completely been understood. Forinstance it is not yet clearwhat thephysical origins of this stress increaseare. In addition, the influence of the internalmicro structure of concrete,and the mechanisms which lead to different crack patterns whenvarying the loading rate, remain open questions. Unfortunatelyperforming a proper dynamic experimental test in direct tension bringssomedifficulties. One of thebiggest limitations is how toattach properlythe specimen to the testing machine. Therefore, some authors have

used, for high strain rates, an indirectmethod employing theHopkinsonbar [33,3]. In this case, a compressivewave, sent to the specimen, travelsthrough the sample until it finds the free boundary and reflects backalready as a tensile wave. Then, the specimen breaks by spalling.However, it is still not clear whether or not the incident compressivewave could have affected the material before the tensile wave initiatesits way back.

Concrete is a heterogeneous material made of cement (8–15%),water (2–5%), and about 80% of aggregates including gravel, sand,limestone filler and finally other chemical additives in proportionssmaller than 0.5%. The result of thismixture is a quasi-brittlematerial inwhich, the wide kaleidoscope of sizes of the ingredients defines themechanical behavior. Although all the ingredients influence the finalmechanical behavior, it seems to be that every ingredient has aninfluence according to its characteristic length. In this sense, andaccording to the levels of observation proposed byWittman [42], meso-scale (m−3) andmicro-scale (m−6) should be the logicalworking scalesfor material scientists in concrete research. Nevertheless, micro-scale isstill far from the computational capabilities if ‘real’ size specimenswantto be analyzed. Thus, different meso-scale models have been tested byseveral authors for the simulation of concrete behavior and fracture. Afirst class of models has recourse to the finite-element approach, inwhich concrete is represented either as a mixture of aggregates in amatrix with an interfacial transition zone between them or using adamage formulation [29,41,6,7,39,24,19]. A second class to simulateconcrete behavior involves lattice models [35,2], where the mechanicalproperties of the lattice beams aim to represent the concrete meso-structure [17,12]. Thesemodels have demonstrated success at capturing

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themain characteristics of the overall mechanical behavior of concrete:crack initiation, propagation, coalescence and localization.Moreover theconstitutive equationsmight be relaxedwhen comparingwith amacro-scale formulation, thereby reducing the number of model parameters.

This paper aims at exploiting the capabilities of a 2D (finite-element)meso-mechanical model of concrete. The dynamic propagating crackswill bemodeled explicitly byhaving recourse to the relatively-little used(in concrete modeling) cohesive-element approach. The objectives ofourwork are twofold. First,wewish tomake explicit the influenceof thetoughness of aggregates on the macroscopic response. To this end, weconsider two different types of aggregates differing in their properties(called hard and soft aggregates). The spatial ordering of the meso-structure and the aggregates size are investigated as well. Second, westudy the rate dependence of concrete in relation with its meso-structure composition. We propose to adopt a rate dependent cohesivelaw to reproducewith better accuracy the experimental observations ofstrain-rate strengthening. While rate dependency has been taken intoaccount in several concrete models, using either a rate-dependentdamage formulation [16,9], visco-plasticity [37] or a coupling betweenboth formulations [10], little attempts have been made to couple ratedependency with a meso-scale modeling of concrete. A noteworthyexception includes some recent work, for instance Cusatis et al. [45]within the lattice frameworkand [26]within a (viscoelastic viscoplastic)damage formulation with interfacial transition zone. Our work willextend this line of thought to the finite-element/cohesive-elementapproach.

The paper is decomposed as follow. The next section describes thefinite-elementmodel andmaterial parameters. Section3 constitutes themain part of the paper and gathers a mesh convergence analysis andparametric studies showing the influence of aggregates properties,spatial ordering, sizes, and strain rates. A discussion on the advantage ofa rate-dependent cohesive law formulation is brought in Section 4.

2. Approach

2.1. Meso-structure generation

Our model considers concrete as a material made only of twocomponents. Beyond the matrix phase, only the large aggregates arerepresented explicitly. The small aggregates and other componentsare assumed to be mixed up with the mortar phase establishing thematrix phase. For the mesh generation the large aggregates areinitially represented by means of Voronoï polyhedrons. To generatethem, a program starting from a Voronoï mesher has been developed.

To prevent thefinal Voronoï geometry of presenting tiny edges [48]a Body Centered Cube (BCC) was chosen for the initial distribution ofpoints. The generation points code is linked to the Voronoï generatorprogram and the resulting geometry (in terms of Delaunay mesh) islinked to another codewhich detects each polyhedron, shrinks, movesand rotates it following an uniform random distribution. As a result, aset of Voronoï polyhedrons surrounded by a box is obtained, Fig. 1.

The matrix phase is created automatically using the capabilities ofa mesh generator program by considering the specimen surface

Voronoi Polyhedra SDistribution of points

Fig. 1. Schematics of the generation

(external box) and excluding the Voronoï polyhedrons (aggregates).Notice that this is possible since the cohesive element methodologyhere does not require an a priori definition of the possible fractureplanes, since cracks are constrained to propagate following the inter-element boundaries [29,36,41,7].

With this method, we generated 2D (plane strain) squarespecimens with an external size h of 0.1×0.1 m2.

2.2. Numerical dynamic approach

Our so generated finite element meshes are composed of 6-nodedquadratic triangles. In order to find the equilibrium, we need to solvethe discretized equation of motion

M x→::+

→R

int=

→R

ext ð1Þ

To solve this equation one needs to use a convenient finitedifference expression to approximate the accelerations and velocitiesin terms of displacements. We have chosen the explicit version of thepopular second order Newmark β-method. In this case the massmatrix is lumped and therefore diagonal, which allows to triviallyinverse the mass matrix solving explicitly the scheme.

The presented scheme is stable under the condition that the timestep is below a critical value, which is

tcrit = αminNe

e=1

lec

� �ð2Þ

where c represents the plain strain stress wave speed and le is thedimension of the element. The stable time step has to be chosen equalto the smallest value over all elements multiplied with a securitycoefficient α (typically around 0.1).

2.3. Cohesive zone approach

The finite-element method with cohesive capability allows us tomodel dynamic crack propagation and damage in a brittle material likeconcrete. The fracture process is described by the cohesive approach(introduced by Dugdale [46] and Barenblatt [47]) as a separationprocess occurring at the crack tip in a small region of material calledcohesive zone. In this region, the two faces of the crack separateproducing a displacement jump

→Δ = Δn;Δsð Þ. The debonding process is

described by a constitutive relation between traction and openingdisplacement. The bulk material outside the cohesive zone remainsundamaged and it continues to behave linear elastically.

2.3.1. Cohesive zone insertionDuring the simulation, the stress on the interface between two

adjacent continuum elements, Fig. 2, is computed and checked againstthe selected fracture criterion at the end of every time step.

The interfacial stress, σ, is computed averaging stresses of theadjacentGauss points of the two continuumelements. This interpolation

hrinked Voronoi = aggregates Matrix

of concrete's meso-structure.

Fig. 2. Schematics of the nodal disconnection between two continuum elements inwhich the interfacial stress has exceeded the critical stress.

Fig. 3. Graphical representation of the linear cohesive law.

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of the interfacial stress, although introduces some small error, convergesto the solution as the finite elements become small.

If the inter-element stress exceeds the critical stress value, themiddle node located at the inter-element boundary is doubled, thetwo elements are topologically disconnected and a cohesive elementis inserted.

Therefore a crack can propagate only when a cohesive element isinserted adjacent an already existent one allowing the shared cornernode to be duplicated as well. After the nodal disconnection, theinterfacial stress starts being controlled by the traction separation law,which is implemented in the cohesive element.

2.3.2. Cohesive constitutive modelThe coupling of the above mentioned fracture criterion together

with an initially rigid cohesive law allows capturing the initiation ofnew cracks without an a priori definition of the possible cracks, whichare constrained to propagate following the inter-element boundaries.The law we use is the linear irreversible softening law proposed byCamacho and Ortiz [8,27].

The authors hypothesize the existence of a free potential energy ϕ,which depends not directly on the relative displacements,→Δ = Δn;Δsð Þ, but in an effective scalar displacement which has thefollowing form:

δ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ2

n + β2Δ2s

q: ð3Þ

The parameter β accounts for the coupling between normal andtangential displacement. The value of β has to be estimated (e.g. bycorrelating experimental results with numerical simulations, [34]),and is therefore bound with some uncertainty. We investigated theinfluence of that parameter by varying its value between zero (onlynormal tractions) and 10. We could establish that the peak strength isonly slightly affected by β, as we are applying a tensile loading. For theremainder of the paper we will keep its value equal to one. The freeenergy function has the form in the reversible case

ϕ δð Þ = 12fctδ 2− δ

δc

� �ð4Þ

where fct represents the local material strength and δc represents theeffective relative displacement beyond which complete decohesionoccurs. The derivation of the free potential energy with respect to theopening displacement leads to the cohesive tractions:

→t =

∂ϕ∂→Δ

=tδ

Δn→n + β2→Δs

� �ð5Þ

where t represents an effective cohesive traction. This traction in caseof crack opening is given by:

t = fct 1− δδc

� �for δ = δmax and δ̇ > 0: ð6Þ

Whereas for crack closure or reopening (δ smaller than δmax) thefunctional form is assumed to have the form:

t = tmaxδ

δmaxfor δ < δmax ð7Þ

where tmax is the value of the effective traction when δ is equal to δmax,in which the maximal effective opening displacement attained up tothe moment is stored. Moreover δmax also accounts for theirreversibility of the law allowing successive loading, unloading andreloading. The evolution of the linear decreasing law is graphicallyshown in Fig. 3.

Note that the definition of fct and δc implicitly establishes theexistence of an effective fracture energy Gc, which corresponds to thearea under the curve of Fig. 3:

Gc =12fctδc: ð8Þ

Partially damaged cohesive elements have dissipated an energyW.

2.4. Material parameters

Meso-mechanical models need to be fed with specific materialparameters, which may require specific experiments. In particular,fracture properties of the components (aggregate, matrix, whichincludes mortar and small aggregates, and aggregate-matrix inter-face) are needed.

In real concrete, the value of those parameters influences stronglythe macroscopic mechanical behavior. In particular, the strength andstiffness of large aggregates (with respect to the matrix's ones) and thestrength of the aggregate-matrix interface are directed correlated withthe final experimental results, i.e. cracking patterns, fracture energy,critical crack opening, softening curve and Young's modulus [11].

This work only considers mean values of those parameters.Nevertheless two different types of aggregates have been considered.In Table 1, a summary of the chosen material parameters is given.

In order to validate our model, we have performed two simulationsusing the same specimen geometry of the quasi-static tests presented inone of the references [31] from which the aggregate and matrixproperties have been taken. The numerical results lie within theexperimental scatter and were therefore in good agreement with theexperimental load displacement behavior.

Table 1Summary of fracture material properties taken from [32,31].

Material properties Density (kg/m3) E (GPa) ν Gc (J/m2) fct (MPa)

Hard aggregate 2700a 19 0.2 60 16Soft aggregate 2700a 2.1 0.2 20 1.7Matrix 2200a 31 0.2 52 4.7Aggretate-matrix – – – 30 4.7b

a These values are generic and chosen from Caballero [48].b This is an estimated value, set equal to the matrix–matrix interface strength, as

direct experimental data was not found.

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2.5. Measure of damage

When using continuum damage models, damage is defined as avariable at the constitutive level. The visualization of this variable cantherefore give an approximate idea about the irreversible deformationof a structure and the underlying damage. On the contrary, with thecohesive element approach, cracks are modeled explicitly, giving anopportunity to construct relevant damage indicators and to monitortheir time evolution. To this effect, we defined the following variable.

2.5.1. Damage normalized fracture work (DNFW)The index accounts for the ratio between the work fracture spent

(already dissipated fracture energy W) and the fracture energynecessary to split the interior specimen in two, i.e. it is the fractureenergy corresponding to a length equal to h, which is the specimenwidth (this allows to have an index insensitive to the mesh size).

DNFW =

∑nbinterf aces

i=1∫li

Wi dli

Gc⋅hð9Þ

Where li is the length of cohesive element I, Wi the partiallydissipated cohesive energy and Gcrepresents a mean fracture energy,weighted with respect to the amount of fracture energy potentiallyreleased by each inter-element component.

In order to distinguish and to localize damage, the above damagevariable has been also defined for each material phase: aggregates,matrix and aggregate matrix, and can be indentified (as for the othervariables) with a corresponding subscript: a, a–m and m.

3. Results

3.1. Geometry and boundary conditions

All the specimens are loaded under displacement control. To avoidstress wave propagation from the boundaries and early fracture nearthe boundaries, all the nodes computations are given an initialvelocity, [23], see Fig. 4.

Fig. 4. Boundary (left) and initial (right) conditions.

Nodes of the finite element mesh which are located on the upperand lower boundaries are forced tomove at a certain constant velocityv0:

v0 = ε̇h2

ð10Þ

where ε̇ is the strain rate and h the height of the specimen. The upperboundary respectively the lower one moves upward respectivelydownward at velocity v0. To avoid stress wave propagation from theboundaries and early fracture near the boundaries, all the nodescomputations are given an initial velocity in respect to their verticalposition y [23] as illustrated in Fig. 4.

vy yð Þ = 2v0h

y ð11Þ

In the following subsections the work is focused first on theconvergence of the simulations (Section 3.2); later an investigation onthe influence of the aggregates toughness (Section 3.3) and theirordering (Section 3.4) and size (Section 3.5) is given.

3.2. Effect of mesh size, geometry and material parameters

The fracture process in brittle solids is usually assumed to be atoughness controlled phenomenon, which means that the process iscontrolled by Gc and is therefore independent of fct of the cohesivezone.

Conversely, if the macroscopic strength is linked with fct, theprocess is said to be strength controlled. This must be valid not onlytheoretically but also when running a numerical simulation, whichuses the cohesive zone approach.

It is well known that the relationship between the cohesive lawand the critical energy release rate Gc introduces a length scale intothe material description. This imposes a constraint on the mesh size.This length scale, calculated by Rice and Palmer [28,25], is calledcohesive zone length lz and is defined as follows for Mode I crackloading in case of plane strain

lz =932

πEGc

1−v2� �

f 2ct: ð12Þ

This length has an important influence around the crack tip innumerical simulation. In general, for toughness-controlled fracture,the following relation has to be satisfied:

lm < lz < h ð13Þ

where lm is the element size (biggest element edge) in the cohesivearea and h is the width of the specimen (0.1 m). This means that thecohesive zone must contains several elements (typically around four)and that the cohesive zone should be small in comparison to thespecimen size. Table 2 shows the calculated values of the cohesivezone length for the different components of concrete.

Note that it is not possible to clearly calculate a cohesive zonelength for the interfaces between aggregates and matrix (because thechoice of Young's modulus is ambiguous). The cohesive zone lengthfor the hard aggregates and that one between soft aggregate and

Table 2Cohesive zone lengths lz [mm] for the different components of concrete meso-scalemodel. For Aggregate-Matrix interfaces: EM means that Matrix Young's Modulus isconsidered, EHA and ESA this of hard respectively soft aggregates.

Material HardAggregate

SoftAggregate

Matrix Aggregate-matrix

EM EHA ESA

Cohesive zone length [mm] 4.1 13.4 67.2 38.8 23.7 2.6

Fig. 5. Macroscopic peak strength σmax [MPa] versus average element size lm [mm] for a) ˙ε=10s−1 and b) ˙ε=1000 s−1 (SA).

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matrix have the smallest values; therefore they will determine anupper limit for the size of the elements when generating the mesh.

In order to verify the relationship between the different lengthscales a series of numerical tests was carried out. A first test wassimply to load the 0.1×0.1 m concrete specimen varying the degree ofrefinement of the mesh in order to verify the mesh sensitivity. We ranseveral simulations for concrete with soft and hard inclusions fordifferent loading rates. The results for two different loading rates andconcrete with soft aggregates are reported in Fig. 5.

From Fig. 5a) it is possible to observe that there is convergence foran average element length (lm) smaller than about 0.25 mm.Considering this length and looking back at Table 2 we can remarkthat lm should be at least roughly ten times smaller than the Rice'scohesive zone length of the interface, which was derived for pre-cracked specimen with homogeneous materials and without meso-structure. Most published work having recourse to the cohesive-element framework uses an estimate of lm≤ lz/3. Our work demon-strates the importance of using a more conservative estimate in theabsence of a complete mesh-convergence study. The scenario issimilar at high strain rates (e.g. beyond 100 s−1), Fig. 5b), althoughthe peak strength is higher and apparently a finer mesh is needed toreach a fully converged results. The chosen mesh-size for thesimulations presented in this paper is set to 0.4 mm. Although this

Fig. 6. a) Mesh of the pre-cracked specimen. b) Evolution of the macroscopic peak strength σcohesive length as for soft respectively hard aggregates.

value seems to be related with some limited mesh dependency, itallowed us to save some computational time.

A closer inspection of Table 2 reveals yet another challenge. Thematrix cohesive length has roughly the same dimension of the testedspecimen; therefore the relation of Eq. (12) is not respected. Wecarried out a second study in order to highlight the influence of thissecond requirement on the macroscopic strength. A pre-crackedspecimen (Fig. 6a)), made only of one component, has been chosen forthis test. The specimen is loaded with a strain rate of 10 s−1, thehorizontal displacements of the left-boundary nodes were blockedand thematrix-component was selected as referencematerial. We randifferent calculations keeping Gc constant (52 J/m2) and varying thecritical opening strength fct between 1.175 Mpa (0.25∙fct,m) and75.2 Mpa (16∙fct,m). According to the previous investigation ofconvergence, the mesh has been refined in the fracture area, wherethe elements have an average edge length of 0.5 mm.

The results are displayed in Fig. 6b). One can distinguish threedomains. On thefirst oneason the third one, themacroscopic strength isclearly an increasing function of fct, while region two is staying quiteconstant. The domain two is clearly the toughness-controlled one,whereas domain one and three are strength-controlled.

On domain one, lz is bigger than 0.01 m, which is a too big valuecompared to the specimen size h (0.1 m). On the third domain the

max varying fct. The square and triangular symbols are located in the plot with the same

Fig. 7. Pre-cracked specimen with variable width.

Fig. 9. Concrete meso-geometry with 79 aggregates (left). Zoom of the correspondingfinite-element mesh (right).

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cohesive zone becomes too small compared to the size of the chosenmesh (lzb2.5 mm), which means that the mesh is not fine enough tocapture the cohesive zone. The cohesive lengths for matrix (roundsymbol) and soft aggregates (square symbol) are included in domainone. Whereas, for hard aggregates (triangular symbol), the cohesivelength is located in the toughness-controlled region (domain two).

We further carried two tests to quantify the influence of thechosen specimen size on the peak strength. Both tests are conductedfor a fixed mesh size. One consists in taking a specimen with a regularmeso-structure and the standard geometry tested in this paper(0.1×0.1 m), but with a pre-crack. The specimen width has beenincreased by keeping the height constant (Fig. 7).

We could observe (Fig. 8a)) that the peak strength decreases withincreasing h and converges to a fixed value for a width ofapproximately 0.6 m (which corresponds roughly to ten times thecohesive zone length for the matrix).

The other test consists in enlarging the size of the specimenkeeping the same degree of mesh refinement (Fig. 8b)). We have runsimulations with double and quadruple specimen size. In this case thesize does not seem to affect the peak strength (only a small variationin order of two percent could be observed). When comparing theresults of the two specimens, one can remark that, if there is a pre-crack, the size of the specimen clearly affects the peak strength as thecohesive length of the matrix is in the same order as the chosenspecimen width. Whereas, if the sample is not pre-cracked, it seemsthat the chosen size has not the same influence (as long as the chosenspecimen size can be considered a representative volume element).

For the chosen specimen geometry, and material parameters, thepresented results are not in a well-defined region. We have tried toreduce as much as possible the mesh dependency, but this iscomplicated by the presence of a meso-structure. The concept oftoughness-controlled regime applies well for homogeneous materialsand pre-cracked specimen. However, in our case, where multiplecracks are opening (during the simulations) and interacting, theupper limit for the specimen size of Eq. (13) has to be relaxed.

Fig. 8. a) Macroscopic normalized peak strength σmax versus specim

Moreover, because of the meso-structure, which is composed ofdifferent materials, is not possible to clearly define a cohesive zonelength. Without a rigorous estimate, it was decided to use a mesh sizesmaller than a sixth than the smallest cohesive zone estimate.

Thus, it can be concluded that, for the geometry of the concretespecimens presented in this paper, the present work lies in a mixedstrength-controlled and toughness-controlled regime.

3.3. Soft vs. Hard aggregates and strain rate effect

To study the influence of themechanical properties of the aggregateson the behavior of concrete, two different types of aggregates, differingin their material properties, have been considered: soft and hard (seeTable 1). The concrete meso-structure is modeled with the geometryand mesh presented in Fig. 9.

The response of the two types of concrete depends on the strainrate at which the specimen is loaded. For instance, two stress-straindiagrams are plotted in Fig. 10a) for a low strain rate ( ε̇ = 10s−1), b)for a high strain rate ( ε̇ = 1000s−1).

From Fig. 10a) one can recognize that the aggregate typeinfluences the response of the concrete specimen only at low strainrates. As mentioned previously, we keep the value of β, the couplingparameter in the cohesive law, equal to 1. Since we are in mode I, thisparameter does not influence significantly the peak strength (a studyof the influence of β was conducted but is not shown here). Thevariation of peak strength with strain rate for the two differentaggregates is highlighted in Fig. 11a). The results display a strain ratehardening. However, the computed increase of strength is quitesmaller than the experimental one see Fig. 11b). We will go back to

en with h [m]. b) Specimens with double and quadruple size.

Fig. 10. Macroscopic stress-strain (σ−ε) comparison between hard and soft aggregates: a) at low strain rate ( ε̇=10s−1) and b) at high strain rate ( ε̇=1000 s−1).

1136 L. Snozzi et al. / Cement and Concrete Research 41 (2011) 1130–1142

this point in Section 4, where we motivate the use of a phenomeno-logical rate dependent cohesive law.

Under a low strain rate, the presence of soft aggregates withinconcrete makes the material softer in its pre-peak behavior. However,this effect starts to disappear when the strain rate is increased above10 s−1. The peak strength is also decreased by the presence of softaggregates when concrete is loaded under low strain rates (the peakstrength for hard aggregates is approximately double than for the softones). Finally, at low ε̇ the softening branch shows a more attenuateddecay when comparing soft vs. hard aggregates.

This scenario changes completely when analyzing the samespecimens under high strain rate, Fig. 10b). The results show thatthere are no differences between soft and hard aggregates in terms ofstress–strain diagrams. The stress–strain plots are, almost identical,exhibiting same elastic slope, peak strength and softening decay up tothe last computed time.

The same distinction between the behavior at low and high strainrates can be done when looking at the deformed mesh configuration,which is plotted in Fig. 12 for the two different types of aggregates.

One can remark in the case of hard aggregates, that at low strainrates some macro-cracks propagate in the matrix avoiding theaggregates, whereas the cracks cross the aggregates if soft inclusionsare chosen. Therefore the failure process is dominated by crack

Fig. 11. a) Macroscopic peak strength σ [MPa] — strain rate ε̇ [s−1] for soft and hard aggrefollowing experimental data: [5,30,1,38,21,15] (note that the vertical axis is normalized by

propagation. By increasing the loading rate cracks start to nucleate inthe matrix and have less time to propagate before the peak strength isreached. At high strain rate (1000 s−1) the crack path for soft and hardaggregates is the same. This means that the peak strength isdominated by crack initiation.

This phenomenon can also be highlighted when considering thedamage normalized fracture work.

Fig. 13 shows the damage diagrams for each of the differentconcrete phases: aggregates, matrix and aggregate-matrix interfacesrespectively. In all figures, damage is plotted against the total tensilestrain applied to the specimen, which is measured as the totalelongation divided by the original length of the specimen. Softaggregates (SA) are compared vs. hard aggregates (HA) under both:a) low and b) high strain rates.

As noticed from the deformed mesh configuration, under lowstrain rate hard aggregates do not experience any damage, whereasthe damage on soft aggregates increases with increasing strain level.For both aggregates the damage in the matrix appears first. This iscongruent with the fact that at early stages of loading, i.e. before peakstrength, a large amount of micro cracks starts to be nucleated withinthe matrix. With increasing strain those micro-cracks begin topropagate and therefore damage outside from the matrix phase canbegin.

gates, b) Comparison with experimental values extract from [4], which itself has usedthe static strength).

Fig. 12. Crack propagation in the mesh: at low strain rate for hard a) and soft b) aggregates, at high strain rate c) just one zoom is showed (crack nucleation occurs everywhere in thematrix independently of aggregate type).

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When a tensile wave traveling throughout the matrix reaches anaggregate, it induces a tensile strain on it. However, the deformationthat the aggregate will experience will depend on the Young modulusratio between the matrix and the aggregate. Thereby, the stiffer is theaggregate with respect to the matrix, the lower the deformation onthe aggregate will be. This implies a larger opening of the interface.Therefore, the introduction of aggregates much harder than thesurrounding matrix will concentrate the damage on the interfacesrather than in the inclusions as in the case of soft aggregates.

Damage in the matrix for hard aggregates is also higher whencompared with soft aggregates. This agrees with a larger value ofdissipated fracture energy and higher peak strength. After the peakstrength has been reached and the softening behavior starts to bemore accentuated the rate of fracture energy dissipation decreases forthe damage indicators of the three components.

At high strain rates the damage is localized mainly in the matrix.The high loading rate does not allow cracks to propagate before thepeak strength has been reached. Micro cracks start to nucleateeverywhere in the matrix, which displays the highest stress wavespeed and therefore crack propagation speed too. This more diffusemicro-cracking is related to the intrinsic characteristic opening timeof the cohesive law (derived in Camacho et al. [8]). Therefore, for ourcomputations, we cannot explain the increase of strength with a moretransgranular fracture [43].

Interesting, the dissipated fracture energy at peak strength seemsto be only slightly affected by the loading rate. However, post-peakdissipation of energy rises with increasing strain rate and is clearly anincreasing function of the loading velocity.

The presence of two different types of inclusions not only affectsthe damage value but also the crack pattern and its evolution. To thisfact we consider the variable, Nclusters, which accounts for the number

Fig. 13. Damage on aggregates, matrix and interfaces vs. tensile–strain. Comparis

of “clusters” present in the specimen. A cluster is a group of cracks (i.e.a group of fully broken interfaces for which the opening of thecohesive elements is beyond δc), which are interconnected to eachother.

In general there is a relation between strain rate and when theclusters appear. Relatively, a higher strain rate delays the formation ofthe first clusters. At low strain rates (10 s−1), the first clusters appearfor soft aggregates just before the peak strength has been reached,whereas for hard aggregates they appear immediately after (theprocess being more energy consuming). With increasing strain rate,the clusters will appear only when a bigger amount of fracture energyhas been spent and the peak strength will be reached with a morediffuse nucleation of micro-cracks. Therefore, unlike for ceramicswhere the material breaks down when a cluster has reached a lengthcorresponding approximately to the specimen size [14], it seems thatit is not possible to find a critical cluster length to be related with thefailure of the material.

3.4. Orderered vs disordered meso-structure

This section discusses the effect that themeso-structure has on thefinal fracturing behavior of concrete. The main target is to evaluatewhat is the influence of the location of aggregates and their externalshape. Three different meso-structures are considered:

• Fully uniform, where all the aggregates have the same shape(excepted those located near to the borders due to the Voronoïgeneration process), same orientation and are uniformly distribut-ed. This distribution will be referred as U.

• Fully uniform plus random rotation, where as in the previousdistribution all the aggregates have same external shape but they

on between soft and hard aggregates: a) low strain rate, b) high strain rate.

Fig. 14. Different meso-structures: a) uniform aggregates (U), b) rotated aggregates (URot) and c) completely random (R).

1138 L. Snozzi et al. / Cement and Concrete Research 41 (2011) 1130–1142

are randomly rotated with respect to their own center. Thisdistribution will be referred as URot.

• Fully random, aggregate's shape and position are randomlyassigned. This distribution will be referred as R.

Fig. 14 shows the three different meso-structures considered. Allthe meso-structures have the same ratio of aggregate volume, 30%,although due to the randomness different number of aggregates. Thus,U and Rot meso-structures have 67 aggregates of approximately8.5 mm whereas R distribution has 79 aggregates, where mostaggregates have sizes between 6 and 10 mm. All the computationshave been carried out considering particles as hard aggregates, sincethe influence of the meso-structure is not significant for the softinclusions (which are weaker than matrix grains interfaces).

Fig. 15 plots the stress–strain diagrams for the different meso-structures under low (a) and high (b) strain rate. From Fig. 15 we canextract the following conclusions.

The initial elastic slope is not affected by the meso-structure.The peak strength is always lower for the U distribution. This is

probably due to the fact that, in the U distribution, all the aggregateshave two faces orthogonally aligned to the loading direction, whichreduces the maximum strength. Indeed, considering the dissipatedfracture energy of the interfaces (Fig. 16) for the U specimen, one canremark that it is larger than those of the other two meso-structures(URot and R). On the contrary, the dissipated fracture energy of thematrix is smaller for the U specimen.

The preferential direction of failure is avoided in the URot and Rdistributions. Comparing URot and R specimens, the fully disorderedmeso-structure increases the possibility of having weak zones (grain-matrix interfaces are closer in some places). In such regions, damagemight grow faster and therefore the global mechanical responsesoftens earlier.

Fig. 15. Stress–strain diagrams obtained for the three differe

Finally, material softening is slightly more attenuated in the Rdistribution. As mentioned before, the R distribution contains “weakzones”, which facilitates the initiation of cracks and their local growth.However, after an initial growthandnucleation of cracks, the disorderedmeso-structure may obstruct the further propagation of cracks (we areunder the assumption of hard aggregates), and forces the cracks todeviate around the aggregates, which requires more energy (thedissipated fracture energy in the matrix of the R distribution is slightlyhigher compared with the one of URot distribution).

3.5. Influence of aggregate's size

In this section the effect of the aggregate's size is analyzed. Threedifferent specimens with the same standard geometry of0.10×0.10 m2 are considered. In all specimens, volume ratio ofaggregates is kept constant and equal to 30%, whereas the size of theaggregate is decreased as the number of inclusions is increased. Theaverage aggregate size of each specimen is: 8.5 mm, 3.9 mm and2 mm (corresponding to a number of aggregates of 66, 297 and 1198).The three different specimens considered are shown in Fig. 17. Bothtypes of aggregates, soft and hard, are considered.

Fig. 18 shows the obtained stress–strain diagrams for the differentconcrete specimens under low and high strain rates. At both loadingrates the elastic modulus appears to be independent of the aggregatesize. At low loading speed the peak strength for hard aggregates is notsignificantly affected by the number of inclusions (not shown infigure). Whereas, for soft aggregates, it is possible to notice anincrease in peak strength with increasing number of aggregates(Fig. 18a).

In a concrete material with large aggregates (which representweak zones in the case of soft aggregates) cracks can nucleate beforeand propagate easier within the inclusions than in a concrete with

nt meso-structures under a) low and b) high strain rate.

Fig. 16. Dissipated fracture energy for the matrix (in green) and interfaces (in red) vs.tensile strain. Comparison between specimens with the three different meso-structures.

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small aggregates. Indeed, cracks nucleate first in the matrix and thenpropagate from the matrix in the aggregates too. Therefore, if theaggregates are bigger, the cracks will cross them easier breaking alarger area of aggregates leading to lower peak strength.

This phenomenon can also be observed when looking at the graphfor the dissipated fracture energy (Fig. 19a)). The dissipated energy inthe matrix increases with increasing number of aggregates. Incontrast, the aggregate's dissipated energy reduces with decreasingsize of inclusions.

Fig. 17. Specimens with different mean size of aggregates: a)

Fig. 18. Stress–strain diagrams obtained for the different

The scenario changes when considering high strain rates(Fig. 18b)). In this case the peak strength is higher for large inclusions(independently of aggregate type). We recall that at high strain ratethe cracks nucleate in the matrix and aggregate-matrix interfacesleading to a similar stress–strain behavior for the two types ofinclusions. Therefore, even though aggregates seemnot be directly thecause of the variation of the strength, a larger number of aggregatesimplies also a higher probability for a growing crack tofind an interfacebetween mortar and aggregates (interfaces are weaker than matrix).

As shown in Table 3, for the same volume ratio of inclusions, thetotal length of the interfaces (between aggregates and mortar)increases roughly four times between the 66 and 1198 aggregatesspecimens. This is congruent with a decreasing ratio of dissipatedfracture energy at the interfaces with increasing aggregates size(Fig. 19b)).

4. A rate-dependent cohesive law

In this section we discuss the limit of the proposed approach.Although, the presented meso-mechanical model is bound with acertain freedom, there are also some limitations. Indeed, (as noticed inthe results section) when comparing our results with experimentalones, it is possible to remark that the rate-independent cohesive lawdoes not allow to have a strain-rate hardening comparable with theexperimental one. Moreover the dissipated fracture energy at peak inthe simulations seems to be rougly independent of strain rate(however post peak dissipation of energy is clearly an increasingfunction of strain rate).

Whereas, experimental results have shown that energy dissipationincreases with the loading rate [33,3]. Although, there is a generalagreement on this phenomenological behavior, it is still not clear

66 aggregates, b) 297 aggregates and c) 1198 aggregates.

aggregate sizes under a) low and b) high strain rate.

Fig. 19. Dissipated fracture energy vs. tensile–strain. Comparison between specimens with 66, 297 and 1198 aggregate: a) at low strain rate and b) at high strain rate.

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which should be the mechanisms or physics which govern it. Theincrease is attributed to two different aspects by different authors,structural or material. According to the structural effect hypothesis,the explanation resides in a more diffuse micro-cracking, whichshould be linked with higher fracture energy [40,3]. On the contrary,the material explanations sustains that the increase of toughnesscould be an effect of internal material “hardening” [20]. We argue thatthis mechanism should be related to the chosen modeling scale. I.e.when considering a large modeling scale (macro-scale), the tough-ness parameter of the cohesive approach could be a function of theloading rate, as the mesh size cannot be small enough to capture allmicro-cracking mechanism. Whereas, decreasing the scale shouldentail a rate independent toughness parameter.

Nevertheless, the small computed amount of dissipated fractureenergy (the dissipated fracture energy at peak strength seems to beroughly independent of strain rate) and the small increase in strengthwith increasing loading rate in comparison to experiments seems tocall for the introduction of a phenomenological rate-dependentcohesive law. Therefore, we decided to slightly modify the linear-decaying cohesive law in order to take into account the strain rateeffect. Among the various possible laws we identified two differentones differing in their shape and which are also function of differentrate-dependent parameters.

The first variant [44] assumes that the opening cohesive strength isa material constant value, while the critical opening displacement is afunction of the crack opening rate.

Therefore the linear cohesive law is enriched with the followingrelationship for the rate-behavior of the critical opening:

δc = δco 1 +δ̇

δ̇o

!ð14Þ

where δco corresponds to the critical crack opening displacement forthe rate-independent law (static value) and ˙δo represents a scalingcrack opening rate. The introduction of this rate-dependent crackopening increases the value at which complete separation of thecohesive zone occurs with increasing opening rate. This should alsolead to a higher dissipated fracture energy, which could be related

Table 3Interface length between aggregates and matrix for the three different specimens.

Aggregate's number 66 297 1198

Interface length [m] 1.69 3.58 7.17

with a more diffuse micro-cracking that the actual meso-scale is notable to capture.

We ran several simulations with this modified cohesive law fordifferent loading rates in order to verify the strain rate-dependence.Surprisingly, we could not notice a remarkable difference in the peakstrength values and dissipated fracture energy with the rate-independent law. This may be related with the chosen sample's size,for which one we are in presence of a mixed strength and toughnesscontrolled regime.

Therefore, we decided to use a second variant where both σc and δcare sensitive to the loading rate. This results in a law, which expandsoutwards with increasing loading rate as shown in Fig. 20a). Theinconvenient with this kind of law is that the cohesive strength can beseveral times higher than the macroscopic material strength, which isphysically questionable. Moreover, because we are assuming an initialrigid behavior (explicit approach), the cohesive strength needs to beknown a priori. Therefore, we decided to link the law directly with themacroscopic loading ratio, introducing the following strain ratedependencies:

fct = fco 1 +ε̇

ε̇o

!ð15aÞ

δc = δco 1 +ε̇

ε̇o

!ð15bÞ

where δco and fco correspond to the static values of cohesive strengthand critical opening displacement respectively. Thus the two valuesare increased of the same amount with increasing loading rate(Fig. 20b)).

Thanks to the simple loading condition and geometry of thepresented specimens, this second variant has been implementedspecifically for the proposed virtual experimental setup. For morecomplicated loading conditions, it would be necessary to bound thestrain-rate dependence with the deformation rate of the four gausspoints adjacent to the inter-element boundary. Another possibilitywould be to take the static critical stress for insertion and to add,within the cohesive law, a rate-dependent viscosity parameter whichdepends on the opening rate (for instance [18]). A different approach,which would not involve a cohesive-approach, would consist inhaving a visco-plastic bulk material (such as proposed in combinationwith a continuum damage formulation in [13]).

As for the previous variant, we ran several simulations varying theloading rate. The resulting peak strengths are depicted in the graph of

Fig. 20. Two rate dependent cohesive law: a) generalized cohesive lawwith crack opening displacement as a function of crack opening rate, b) cohesive stress and crack opening areboth function of strain rate.

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Fig. 21a). One can remark that with this law the increase in peakstrength and dissipated fracture energy is much more pronouncedand is therefore closer to the experimental values (Fig. 21b)) and abetter agreement can be simply reached by calibrating with theexperimental values.

The fit could be perfected by choosing a non-linear relationship forσc and δc (Eqs. (15a) and (15b)).

5. Conclusions

In this paper we have developed a 2D cohesive zone meso-mechanical and dynamic model of concrete to study its fracturingbehavior under low and high strain rate loadings. Although it is hardto operate in a strict toughness controlled regime, the meso-scalemodel has the significant advantage to represent aggregates andmortar explicitly. Both continuum phases are considered to behaveelastically while initiation, coalescence and propagation of cracks areexplicitly modeled by an automatic insertion of interface elements.We have investigated the influence of the aggregate's properties (e.g.referred to soft and hard aggregates, which differ in toughness andstrength, but have a narrow range of elastic stiffness), the effect of theinternal meso-structure (arrangement and shape of the aggregates)and the influence of their sizes on the simulations. From the obtainedresults we can extract the following salient conclusions.

While the constitutive law used tomodel the interfacial behavior isinsensitive to the loading rate, themodel is able to capture the increasein peak strength and strain at failure with increasing rate of loading.The increase in strength with the strain rate resides in a more diffuse

Fig. 21. a) Macroscopic peak strength σ [MPa] — strain rate ε̇ [s−1] for soft and hard aggrefollowing experimental data [5,30,1,38,21,15].

micro-cracking and is related to the intrinsic characteristic openingtime of the cohesive law. However, the dissipated fracture energy atpeak strength seems to be roughly independent of strain rate(although, post-peak dissipation of energy is clearly an increasingfunction of strain rate). Therefore, for our meso-mechanical approach,we cannot discard the combination of inertial with material ratehardeningmechanisms. The introduction of a rate dependent cohesivelaw increases the difference of the computed peak strength betweenlow and high strain rates, leading to a closer match with experiments.

In addition, simulations carried out for two different types ofaggregates (soft and hard) concluded that the peak strength isconsiderably sensitive to the toughness of the inclusions, but only atlow strain rates. There, the crack patterns for the two differentaggregates are different. However, at high strain rates, micro-cracksnucleate in the matrix and have no time to propagate in theaggregates before peak strength is reached. This lead to a similarstress–strain behavior for the two aggregate types (inertial forcesdominate the overall behavior in terms of strength).

The material parameters used in this study have been taken fromthe recent experimental literature [32,31] without modification. Forinstance one should notice that the stiffness of both types of inclusionsis lower than the one of the matrix paste. Therefore, in future work,we plan to extend the range of stiffness (and strength) of theconstituents in order to calibrate our model with other experimentaldata and to establish the domain of validity of the above observations.Note that increasing the aggregates stiffness (and decreasing theinterface strength) may also lead to a reduction of the (slight) mesh-dependency that the model exhibits at very high strain rates.

gates. b) Comparison with experimental values extract from [4], which itself has used

1142 L. Snozzi et al. / Cement and Concrete Research 41 (2011) 1130–1142

A comparison between three different meso-structures consideredbrings the following remarks. The specimen with uniform meso-structure reduces the maximum strength, because of the alignedaggregate's faces with the loading direction. Since this is avoided byrotating the aggregates, the rotated meso-structure shows always thehigher peak strength. Finally, the random distribution increases theprobability to have weak zones (interfaces between aggregates andmatrix are closer), which reduces the peak strength.

Finally, our results show that, having two concrete materials withsame aggregate volume ratio, the one with smaller inclusions willexhibit a larger strength at low strain rate in case of soft aggregates.Large aggregates facilitate the propagation of cracks within inclusions.This involves a larger area of broken aggregates, which is linkedwith alower strength.

With this work we presented a study of Mode-I fracture inconcrete with a meso-structure. Our aim is to expand the modeltowards an engineering scale. We are now incorporating a contactalgorithm in the code (which is an important improvement of themodel in order to deal with compressive cracking) and our goal wouldbe to be able to simulate failure of a simple reinforced concretestructure (like for instance a punching beam), where aggregates aresupposed to play an important role, once meso-cracks are opened andin frictional contact.

Acknowledgments

This material is based on the work supported by the Swiss NationalFoundation under Grant no. 200021 122046/1.

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